\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -2.0 & -3.00 x_{1} & -2.00 x_{2} & -9.00 x_{3} & -5.00 x_{4} & +  7.00 x_{5} & +  1.00 x_{6} & -9.00 x_{7} & -3.00 x_{8}\\
 x_{10}   &  3.0 & +  4.00 x_{1} & +  3.00 x_{2} & -6.00 x_{3} & -6.00 x_{4} & -6.00 x_{5} & +  3.00 x_{6} & +  8.00 x_{7} & -2.00 x_{8}\\
 x_{11}   &  3.0 & -8.00 x_{1} & +  5.00 x_{2} & -8.00 x_{3} & +  2.00 x_{4} & -8.00 x_{5} &    &   & +  2.00 x_{8}\\
 x_{12}   &  -3.0 & -10.00 x_{1} & -10.00 x_{2} &   & -10.00 x_{4} & + 10.00 x_{5} & +  4.00 x_{6} & -2.00 x_{7} & -3.00 x_{8}\\
 x_{13}   &  3.0 & -3.00 x_{1} & +  9.00 x_{2} & +  6.00 x_{3} & +  1.00 x_{4} & -7.00 x_{5} & -3.00 x_{6} & +  6.00 x_{7} & -5.00 x_{8}\\
 x_{14}   &  -3.0 & -7.00 x_{1} & +  6.00 x_{2} & +  2.00 x_{3} & +  8.00 x_{4} & -5.00 x_{5} & +  8.00 x_{6} &   & -5.00 x_{8}\\
 x_{15}   &  -3.0 & -1.00 x_{1} & -9.00 x_{2} & -8.00 x_{3} & -8.00 x_{4} & +  6.00 x_{5} & +  3.00 x_{6} & -7.00 x_{7} & -9.00 x_{8}\\
 x_{16}   &  2.0 & -2.00 x_{1} & +  6.00 x_{2} & +  5.00 x_{3} & +  5.00 x_{4} & -8.00 x_{5} &   & +  3.00 x_{7} & +  3.00 x_{8}\\
\hline
z    &  0.0 & -1.00 x_{1} & +  4.00 x_{2} & +  2.00 x_{3} & +  5.00 x_{4} & +  5.00 x_{5} & -2.00 x_{6} & -3.00 x_{7} & -3.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,16$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  3.00 y_{9} & -4.00 y_{10} & +  8.00 y_{11} & + 10.00 y_{12} & +  3.00 y_{13} & +  7.00 y_{14} & +  1.00 y_{15} & +  2.00 y_{16}\\
 y_{2}   &  1.0 & +  2.00 y_{9} & -3.00 y_{10} & -5.00 y_{11} & + 10.00 y_{12} & -9.00 y_{13} & -6.00 y_{14} & +  9.00 y_{15} & -6.00 y_{16}\\
 y_{3}   &  1.0 & +  9.00 y_{9} & +  6.00 y_{10} & +  8.00 y_{11} &   & -6.00 y_{13} & -2.00 y_{14} & +  8.00 y_{15} & -5.00 y_{16}\\
 y_{4}   &  1.0 & +  5.00 y_{9} & +  6.00 y_{10} & -2.00 y_{11} & + 10.00 y_{12} & -1.00 y_{13} & -8.00 y_{14} & +  8.00 y_{15} & -5.00 y_{16}\\
 y_{5}   &  1.0 & -7.00 y_{9} & +  6.00 y_{10} & +  8.00 y_{11} & -10.00 y_{12} & +  7.00 y_{13} & +  5.00 y_{14} & -6.00 y_{15} & +  8.00 y_{16}\\
 y_{6}   &  1.0 & -1.00 y_{9} & -3.00 y_{10} &   & -4.00 y_{12} & +  3.00 y_{13} & -8.00 y_{14} & -3.00 y_{15} &   \\
 y_{7}   &  1.0 & +  9.00 y_{9} & -8.00 y_{10} &   & +  2.00 y_{12} & -6.00 y_{13} &   & +  7.00 y_{15} & -3.00 y_{16}\\
 y_{8}   &  1.0 & +  3.00 y_{9} & +  2.00 y_{10} & -2.00 y_{11} & +  3.00 y_{12} & +  5.00 y_{13} & +  5.00 y_{14} & +  9.00 y_{15} & -3.00 y_{16}\\
\hline
z    &  -0 & +  2.00 y_{9} & -3.00 y_{10} & -3.00 y_{11} & +  3.00 y_{12} & -3.00 y_{13} & +  3.00 y_{14} & +  3.00 y_{15} & -2.00 y_{16}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 7 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  96.0 & +  4.00 y_{5} & -325.00 y_{10} & -137.33 y_{11} & -28.67 y_{12} & -22.67 y_{2} & -76.33 y_{6} & -759.67 y_{14} & -166.00 y_{16}\\
 y_{13}   &  21.0 & +  1.00 y_{5} & -72.00 y_{10} & -33.00 y_{11} & -8.00 y_{12} & -5.00 y_{2} & -17.00 y_{6} & -171.00 y_{14} & -38.00 y_{16}\\
 y_{3}   &  71.0 & +  2.00 y_{5} & -222.00 y_{10} & -89.67 y_{11} & -39.33 y_{12} & -16.33 y_{2} & -55.67 y_{6} & -555.33 y_{14} & -119.00 y_{16}\\
 y_{4}   &  160.0 & +  7.00 y_{5} & -534.00 y_{10} & -244.67 y_{11} & -61.33 y_{12} & -37.33 y_{2} & -128.67 y_{6} & -1296.33 y_{14} & -285.00 y_{16}\\
 y_{9}   &  4.0 & -0.00 y_{5} & -12.00 y_{10} & -5.00 y_{11} & -2.00 y_{12} & -1.00 y_{2} & -3.00 y_{6} & -30.00 y_{14} & -6.00 y_{16}\\
 y_{15}   &  20.0 & +  1.00 y_{5} & -69.00 y_{10} & -31.33 y_{11} & -8.67 y_{12} & -4.67 y_{2} & -16.33 y_{6} & -163.67 y_{14} & -36.00 y_{16}\\
 y_{7}   &  51.0 & +  1.00 y_{5} & -167.00 y_{10} & -66.33 y_{11} & -28.67 y_{12} & -11.67 y_{2} & -39.33 y_{6} & -389.67 y_{14} & -81.00 y_{16}\\
 y_{8}   &  298.0 & + 14.00 y_{5} & -1015.00 y_{10} & -464.00 y_{11} & -121.00 y_{12} & -70.00 y_{2} & -241.00 y_{6} & -2413.00 y_{14} & -535.00 y_{16}\\
\hline
z    &  5.0 & -0.00 y_{5} & -18.00 y_{10} & -8.00 y_{11} & -3.00 y_{12} & -1.00 y_{2} & -4.00 y_{6} & -35.00 y_{14} & -8.00 y_{16}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.16573417586e-15 & -4.00 x_{1} & -1.00 x_{13} & -2.00 x_{3} & -7.00 x_{4} & +  0.00 x_{9} & -1.00 x_{15} & -1.00 x_{7} & -14.00 x_{8}\\
 x_{10}   &  18.0 & + 325.00 x_{1} & + 72.00 x_{13} & + 222.00 x_{3} & + 534.00 x_{4} & + 12.00 x_{9} & + 69.00 x_{15} & + 167.00 x_{7} & + 1015.00 x_{8}\\
 x_{11}   &  8.0 & + 137.33 x_{1} & + 33.00 x_{13} & + 89.67 x_{3} & + 244.67 x_{4} & +  5.00 x_{9} & + 31.33 x_{15} & + 66.33 x_{7} & + 464.00 x_{8}\\
 x_{12}   &  3.0 & + 28.67 x_{1} & +  8.00 x_{13} & + 39.33 x_{3} & + 61.33 x_{4} & +  2.00 x_{9} & +  8.67 x_{15} & + 28.67 x_{7} & + 121.00 x_{8}\\
 x_{2}   &  1.0 & + 22.67 x_{1} & +  5.00 x_{13} & + 16.33 x_{3} & + 37.33 x_{4} & +  1.00 x_{9} & +  4.67 x_{15} & + 11.67 x_{7} & + 70.00 x_{8}\\
 x_{6}   &  4.0 & + 76.33 x_{1} & + 17.00 x_{13} & + 55.67 x_{3} & + 128.67 x_{4} & +  3.00 x_{9} & + 16.33 x_{15} & + 39.33 x_{7} & + 241.00 x_{8}\\
 x_{14}   &  35.0 & + 759.67 x_{1} & + 171.00 x_{13} & + 555.33 x_{3} & + 1296.33 x_{4} & + 30.00 x_{9} & + 163.67 x_{15} & + 389.67 x_{7} & + 2413.00 x_{8}\\
 x_{16}   &  8.0 & + 166.00 x_{1} & + 38.00 x_{13} & + 119.00 x_{3} & + 285.00 x_{4} & +  6.00 x_{9} & + 36.00 x_{15} & + 81.00 x_{7} & + 535.00 x_{8}\\
\hline
z    &  -5.0 & -96.00 x_{1} & -21.00 x_{13} & -71.00 x_{3} & -160.00 x_{4} & -4.00 x_{9} & -20.00 x_{15} & -51.00 x_{7} & -298.00 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.16573417586e-15 & -4.00 x_{1} & -1.00 x_{13} & -2.00 x_{3} & -7.00 x_{4} & +  0.00 x_{9} & -1.00 x_{15} & -1.00 x_{7} & -14.00 x_{8}\\
 x_{10}   &  18.0 & + 325.00 x_{1} & + 72.00 x_{13} & + 222.00 x_{3} & + 534.00 x_{4} & + 12.00 x_{9} & + 69.00 x_{15} & + 167.00 x_{7} & + 1015.00 x_{8}\\
 x_{11}   &  8.0 & + 137.33 x_{1} & + 33.00 x_{13} & + 89.67 x_{3} & + 244.67 x_{4} & +  5.00 x_{9} & + 31.33 x_{15} & + 66.33 x_{7} & + 464.00 x_{8}\\
 x_{12}   &  3.0 & + 28.67 x_{1} & +  8.00 x_{13} & + 39.33 x_{3} & + 61.33 x_{4} & +  2.00 x_{9} & +  8.67 x_{15} & + 28.67 x_{7} & + 121.00 x_{8}\\
 x_{2}   &  1.0 & + 22.67 x_{1} & +  5.00 x_{13} & + 16.33 x_{3} & + 37.33 x_{4} & +  1.00 x_{9} & +  4.67 x_{15} & + 11.67 x_{7} & + 70.00 x_{8}\\
 x_{6}   &  4.0 & + 76.33 x_{1} & + 17.00 x_{13} & + 55.67 x_{3} & + 128.67 x_{4} & +  3.00 x_{9} & + 16.33 x_{15} & + 39.33 x_{7} & + 241.00 x_{8}\\
 x_{14}   &  35.0 & + 759.67 x_{1} & + 171.00 x_{13} & + 555.33 x_{3} & + 1296.33 x_{4} & + 30.00 x_{9} & + 163.67 x_{15} & + 389.67 x_{7} & + 2413.00 x_{8}\\
 x_{16}   &  8.0 & + 166.00 x_{1} & + 38.00 x_{13} & + 119.00 x_{3} & + 285.00 x_{4} & +  6.00 x_{9} & + 36.00 x_{15} & + 81.00 x_{7} & + 535.00 x_{8}\\
\hline
z    &  -4.0 & -83.00 x_{1} & -19.00 x_{13} & -54.00 x_{3} & -138.00 x_{4} & -2.00 x_{9} & -19.00 x_{15} & -40.00 x_{7} & -275.00 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.16573417586e-15 & -4.00 x_{1} & -1.00 x_{13} & -2.00 x_{3} & -7.00 x_{4} & +  0.00 x_{9} & -1.00 x_{15} & -1.00 x_{7} & -14.00 x_{8}\\
 x_{10}   &  18.0 & + 325.00 x_{1} & + 72.00 x_{13} & + 222.00 x_{3} & + 534.00 x_{4} & + 12.00 x_{9} & + 69.00 x_{15} & + 167.00 x_{7} & + 1015.00 x_{8}\\
 x_{11}   &  8.0 & + 137.33 x_{1} & + 33.00 x_{13} & + 89.67 x_{3} & + 244.67 x_{4} & +  5.00 x_{9} & + 31.33 x_{15} & + 66.33 x_{7} & + 464.00 x_{8}\\
 x_{12}   &  3.0 & + 28.67 x_{1} & +  8.00 x_{13} & + 39.33 x_{3} & + 61.33 x_{4} & +  2.00 x_{9} & +  8.67 x_{15} & + 28.67 x_{7} & + 121.00 x_{8}\\
 x_{2}   &  1.0 & + 22.67 x_{1} & +  5.00 x_{13} & + 16.33 x_{3} & + 37.33 x_{4} & +  1.00 x_{9} & +  4.67 x_{15} & + 11.67 x_{7} & + 70.00 x_{8}\\
 x_{6}   &  4.0 & + 76.33 x_{1} & + 17.00 x_{13} & + 55.67 x_{3} & + 128.67 x_{4} & +  3.00 x_{9} & + 16.33 x_{15} & + 39.33 x_{7} & + 241.00 x_{8}\\
 x_{14}   &  35.0 & + 759.67 x_{1} & + 171.00 x_{13} & + 555.33 x_{3} & + 1296.33 x_{4} & + 30.00 x_{9} & + 163.67 x_{15} & + 389.67 x_{7} & + 2413.00 x_{8}\\
 x_{16}   &  8.0 & + 166.00 x_{1} & + 38.00 x_{13} & + 119.00 x_{3} & + 285.00 x_{4} & +  6.00 x_{9} & + 36.00 x_{15} & + 81.00 x_{7} & + 535.00 x_{8}\\
\hline
z    &  -4.0 & -83.00 x_{1} & -19.00 x_{13} & -54.00 x_{3} & -138.00 x_{4} & -2.00 x_{9} & -19.00 x_{15} & -40.00 x_{7} & -275.00 x_{8}\\
\end{array}\]
Final Dictionary
Solution:  -4.0
Num Pivots:  1
\end{document}
